P,Q and R work together to complete a piece of work in x days. P and R take 20 days and 30 days respectively to complete the work. Q is faster than R and slower than P. If x is an integer, then how many values can it take ?

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem step by step, we need to analyze the work done by P, Q, and R, and how they relate to each other based on the information given.

### Step 1: Determine the work rates of P and R
- P takes 20 days to complete the work, so P's one-day work = \( \frac{1}{20} \).
- R takes 30 days to complete the work, so R's one-day work = \( \frac{1}{30} \).

### Step 2: Set up the equation for Q's work rate
- Let Q's one-day work be \( \frac{1}{q} \), where \( q \) is the number of days Q takes to complete the work.
- According to the problem, Q is faster than R but slower than P. Therefore, we have:
\[ 30 < q < 20 \]

### Step 3: Calculate the combined work of P, Q, and R
- The combined one-day work of P, Q, and R is:
\[ \text{Total work} = \frac{1}{20} + \frac{1}{q} + \frac{1}{30} \]
- To combine these fractions, we need to find the least common multiple (LCM) of 20 and 30, which is 60. Thus, we rewrite the equation:
\[ \frac{3}{60} + \frac{60}{60q} + \frac{2}{60} = \frac{3 + \frac{60}{q} + 2}{60} = \frac{5 + \frac{60}{q}}{60} \]

### Step 4: Set up the equation for total work done in x days
- If P, Q, and R together complete the work in \( x \) days, then:
\[ x \left( \frac{5 + \frac{60}{q}}{60} \right) = 1 \]
- Rearranging gives:
\[ 5 + \frac{60}{q} = \frac{60}{x} \]

### Step 5: Solve for q
- Rearranging the equation gives:
\[ \frac{60}{q} = \frac{60}{x} - 5 \]
- Multiplying through by \( qx \) gives:
\[ 60x = 60 - 5qx \]
- Rearranging gives:
\[ 5qx = 60 - 60x \]
- Thus:
\[ q = \frac{60 - 60x}{5x} = \frac{12 - 12x}{x} \]

### Step 6: Determine the range for q
- From the earlier condition \( 30 < q < 20 \):
- For \( q > 30 \):
\[ \frac{12 - 12x}{x} > 30 \implies 12 - 12x > 30x \implies 12 > 42x \implies x < \frac{12}{42} = \frac{2}{7} \]
- For \( q < 20 \):
\[ \frac{12 - 12x}{x} < 20 \implies 12 - 12x < 20x \implies 12 < 32x \implies x > \frac{12}{32} = \frac{3}{8} \]

### Step 7: Determine integer values for x
- The integer values of \( x \) must satisfy:
\[ \frac{3}{8} < x < \frac{2}{7} \]
- Since both bounds are less than 1, we find that \( x \) can take values from 1 to 3.

### Conclusion
- The possible integer values for \( x \) are 1, 2, and 3.
- Therefore, the total number of integer values \( x \) can take is **3**.